RIYADH—Saudi Arabia on Saturday accused Iran of not acting like a nation state and said it is considering further measures against its regional rival, as tensions between the two countries escalated further over the kingdom’s execution of a dissident Shiite cleric.

“Iran has to make a decision whether it is a nation state or a revolution. If it’s a nation state, it should act like one,” Saudi Foreign Minister Adel al-Jubeir said following an emergency meeting in Riyadh with his Gulf Arab counterparts to discuss the crisis...

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